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Rozikov Utkir Abdulloevich

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Total publications: 29
Scientific articles: 29
Citations to the author: 216
Cited articles: 27

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This page:670
Abstract pages:3319
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References:194
Rozikov Utkir Abdulloevich

Professor
Doctor of physico-mathematical sciences (2001)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 20.05.1970
Phone: +998 (71) 262 75 44
E-mail: ,
Keywords: dynamical systems, trajectory theory of dinamical systems, nonlinear operators and processes, random walks in random environment, $p$-adic analysis, the Gibbs measures, lattice models of statistical mechanics.quadratic stochastic operators, simplex, trajectory, Volterra and non-Volterra stochastic operators.
UDC: 512.544.23, 517.2, 517.219, 517.958, 517.98, 517.988.52, 517.998, 519.2, 519.21, 519.248, 530.1
MSC: 28d05, 28d15, 22d20, 22d25, 46l35, 60g99, 60b15, 82b20,92b05,97b10

Subject

The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. For any normal subgroup of finite index the distribution of elements of the partition into conjugate classes on the Cayley tree is described. The notion of wood of Cayley trees is introduced and its group representation is discribed. The uniqueness of the translation-invariant Gibbs measure for the antiferromagnetic Potts model with an external field is proved. For $\lambda $-models the existence of three of translation-invariant and uncountable number of nontranslation-invariant Gibbs measurs on Cayley tree are proved. For any $\lambda$ formula of the critical temperature $ T_c (\lambda )$ is found. The existence of two translation invariant and uncountable numbers of the nontranslationally invariant extreme Gibbs measures for the Ising model with the compete interactions is roved and their constructive description is given. For Ising model with several spin values, Potts model and inhomogeneous Ising model sufficicy conditions of extremity of the unordered phase are obtained. For inhomogeneous Ising model it is proved that there exist only three periodic Gibbs measures corresponding to any normal subgroup of finite index and uncountable number of nonperiodic Gibbs measures. For $\lambda$-models it is proved that there exist only periodic Gibbs measures with period 2 corresponding to any normal subgroups of finite index. On wood of Cayley trees for inhomogeneous Ising model the periodic Gibbs measures are constructed. Using these measures for inhomegeneous Ising model on Cayley tree a new class of Gibbs measures is constructed. A random walk in a random environment on a class of metric spaces are defined. For random walks in a periodic random environment in case when for unit time the particle can transpose to finite distance and for random walks in any random environment when for unit time the particle can transpose only to the neighbouring points, sufficient conditions of non reflexivity are discribed. For example, we consider $Z^d$, infinite trees, free groups.

Biography

Graduated from Faculty of Mathematics and Mechanics of Samarkand State University in 1993 (department mathematical analysis). Ph. D.  thesis was defended in 1995. D. Sci. thesis was defended in 2001. A list of my works contains more than 100 titles.

   
Main publications:
  • Ganikhodjaev N. N., Rozikov U. A. Description of periodic extreme Gibbs measures of some lattice model on the Cayley tree // Theor. and Math. Phys. 1997. V. 111, No. 1, p. 480–486.
  • Rozikov U. A. Partition structures of the Cayley tree and applications for describing periodic Gibbs distributions // Theor. and Math. Phys. 1997. V. 112, No. 1, p. 929–933.
  • Rozikov U. A. Description of limit Gibbs measures for $\lambda$-models on the Bethe lattic // Siberan Math. Jour. 1998. V. 39, No. 2, p. 373–380.
  • Rozikov U. A. Description uncountable number of Gibbs measures for inhomogeneous Ising model // Theor. and Math. Phys. 1999, V. 118, No. 1, p. 77–84.
  • Ganikhodjaev N. N., Rozikov U. A. On unordered phases of certain models on the Cayley tree // Sbornik: Math. 1999. V. 190, No. 2, p. 193–203.
  • Rozikov U. A. Random walks in random environments on metric groups // Math. Notes. 2000, V. 67, No. 1, p. 103–108.
  • Ganikhodjaev N. N., Rozikov U. A. On disordered phase in the ferromagnetic Potts model on the Bethe lattice // Osaka Jour. of Math. 2000. V. 37, No. 2, p. 373–383.
  • Mukhamedov F. M., Rozikov U. A. The disordered phase of the inhomogeneous Potts model is extremal on the Cayley tree // Theor. and Math. Phys. 2000, V. 124, No. 3, 1202–1210.

http://www.mathnet.ru/eng/person8647
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Publications in Math-Net.Ru
1. Periodic Gibbs measures for the Potts model on the Cayley tree
U. A. Rozikov, R. M. Khakimov
Teoret. Mat. Fiz., 175:2 (2013),  300–312
2. $p$-Adic Gibbs measures and Markov random fields on countable graphs
U. A. Rozikov, O. N. Khakimov
Teoret. Mat. Fiz., 175:1 (2013),  84–92
3. The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model
U. A. Rozikov, R. M. Khakimov
Teoret. Mat. Fiz., 173:1 (2012),  60–70
4. A polynomial $p$-adic dynamical system
F. M. Mukhamedov, U. A. Rozikov
Teoret. Mat. Fiz., 170:3 (2012),  448–456
5. Nonuniqueness of a Gibbs measure for a model on the Cayley tree
U. A. Rozikov, G. T. Madgoziev
Teoret. Mat. Fiz., 167:2 (2011),  311–322
6. Description of $p$-harmonic functions on the Cayley tree
U. A. Rozikov, F. T. Ishankulov
Teoret. Mat. Fiz., 162:2 (2010),  266–274
7. The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex
U. U. Zhamilov, U. A. Rozikov
Mat. Sb., 200:9 (2009),  81–94
8. Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree
U. A. Rozikov, M. M. Rakhmatullaev
Teoret. Mat. Fiz., 160:3 (2009),  507–516
9. $F$-Quadratic Stochastic Operators
U. A. Rozikov, U. U. Zhamilov
Mat. Zametki, 83:4 (2008),  606–612
10. Fertile HC models with three states on a Cayley tree
U. A. Rozikov, Sh. A. Shoyusupov
Teoret. Mat. Fiz., 156:3 (2008),  412–424
11. Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree
U. A. Rozikov, M. M. Rakhmatullaev
Teoret. Mat. Fiz., 156:2 (2008),  292–302
12. Potts model with competing interactions on the Cayley tree: The contour method
G. I. Botirov, U. A. Rozikov
Teoret. Mat. Fiz., 153:1 (2007),  86–97
13. A description of harmonic functions via properties of the group representation of the Cayley tree
É. P. Normatov, U. A. Rozikov
Mat. Zametki, 79:3 (2006),  434–443
14. Gibbs measures for the SOS model with four states on a Cayley tree
U. A. Rozikov, Sh. A. Shoyusupov
Teoret. Mat. Fiz., 149:1 (2006),  18–31
15. Group representation of the Cayley forest and some of its applications
N. N. Ganikhodzhaev, U. A. Rozikov
Izv. RAN. Ser. Mat., 67:1 (2003),  21–32
16. Periodic Gibbs Measures for the Ising Model with Competing Interactions
Kh. A. Nazarov, U. A. Rozikov
Teoret. Mat. Fiz., 135:3 (2003),  515–523
17. Representability of Trees and Some of Their Applications
U. A. Rozikov
Mat. Zametki, 72:4 (2002),  516–527
18. Gibbs Measures and Markov Random Fields with Association $I$
A. M. Rakhmatullaev, U. A. Rozikov
Mat. Zametki, 72:1 (2002),  94–101
19. $\mathbb {Z}$Existence of a Phase Transition for the Potts $p$-adic Model on the Set $\mathbb {Z}$
N. N. Ganikhodzhaev, F. M. Mukhamedov, U. A. Rozikov
Teoret. Mat. Fiz., 130:3 (2002),  500–507
20. Countably Periodic Gibbs Measures of the Ising Model on the Cayley Tree
U. A. Rozikov
Teoret. Mat. Fiz., 130:1 (2002),  109–118
21. Periodic Gibbs measures of the inhomogeneous Ising model on trees
U. A. Rozikov
Uspekhi Mat. Nauk, 56:1(337) (2001),  175–176
22. Von Neumann algebra corresponding to one phase of the inhomogeneous Potts model on a Cayley tree
F. M. Mukhamedov, U. A. Rozikov
Teoret. Mat. Fiz., 126:2 (2001),  206–213
23. Random walks in random environments of metric groups
U. A. Rozikov
Mat. Zametki, 67:1 (2000),  129–135
24. The disordered phase of the inhomogeneous Potts model is extremal on the Cayley tree
F. M. Mukhamedov, U. A. Rozikov
Teoret. Mat. Fiz., 124:3 (2000),  410–418
25. On unordered phases of certain models on a Cayley tree
N. N. Ganikhodzhaev, U. A. Rozikov
Mat. Sb., 190:2 (1999),  31–42
26. Construction of an uncountable number of limiting Gibbs measures in the inhomogeneous Ising model
U. A. Rozikov
Teoret. Mat. Fiz., 118:1 (1999),  95–104
27. Description of limit Gibbs measures for $\lambda$-models on Bethe lattices
U. A. Rozikov
Sibirsk. Mat. Zh., 39:2 (1998),  427–435
28. Structures of partition of the group representation of the Cayley tree into adjacent classes by finite index normal subgroups and their application for discription of periodic Gibbs distributions
U. A. Rozikov
Teoret. Mat. Fiz., 112:1 (1997),  170–175
29. Discription of periodic extreme Gibbs measures of some lattice models on the Cayley tree
N. N. Ganikhodzhaev, U. A. Rozikov
Teoret. Mat. Fiz., 111:1 (1997),  109–117

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